Mar. 2005Measurement PuzzlesSlide 1 Measurement Puzzles A Lesson in the “Math + Fun!” Series

Mar. 2005 Measurement Puzzles Slide 1

Measurement PuzzlesA Lesson in the “Math + Fun!” Series


About This Presentation

Edition Released Revised Revised

First Mar. 2005

This presentation is part of the “Math + Fun!” series devised by Behrooz Parhami, Professor of Computer Engineering at University of California, Santa Barbara. It was first prepared for special lessons in mathematics at Goleta Family School during the 2003-04 and 2004-05 school years. The slides can be used freely in teaching and in other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami


Measurement for CookingYou have a 3-cup container and a 5-cup container only.The cups are not graduated. How do you measure one cup of sugar? 3

5

Now add a 10-cup container and show how to measure 11 cups of sugar?

102 = 5 – 3 34 = 3 + 3 + 3 – 5 or 1 + 3 56 = 3 + 37 = 5 + 5 – 3 or 2 + 58 = 5 + 39 = 3 + 3 + 310 = 5 + 5Challenge: Continue to 20 cups. 11 = 10 + (3 + 3 – 5)

1 = 3 + 3 – 5


Activity 1: Measuring Sugar

You have a 1-cup, a 3-cup, and a 9-cup container. The cups are not graduated. How do you measure different amounts of sugar, from 1 to 30 cups, using the fewest number of steps? For example, use 3 once instead of 1 three times.

1

3

9

12 = 1 + 1 34 = 3 + 1 5 = 9 – 3 – 1 6 = 6 – 3 7 = 9 – 3 + 18 = 9 – 1 9 10 = 9 + 1

11 =12 =13 =14 =15 =16 =17 = 18 = 19 =20 =

21 =22 =23 =24 =25 =

26 =27 =28 =29 =30 =


How to Get Half a Glass of Water?

A cylindrical glass is filled to the brim. How do you get half a glass of its content without using any other item?

Full glass Half full

Can’t do this if the glass has a conical, rounded, or irregular shape.

You have a container, with a tight lid or cork, filled to the brim with fruit juice. How do you divide the juice exactly in half without any measuring tools?

Hint: Suppose the container is exactly half full. What happens if we close the lid or cork and turn it upside down?


Measurement at the Hardware Store

A large container is known to hold 24 oz of nails.We have a balance, but no weights.Measure out 9 oz of nails for a customer.

24

12 oz 12 oz

Divide all nails into two equal piles

12 oz 6 oz 6 oz

Divide one pile into two equal piles

12 oz 6 oz 3 oz 3 oz

. . . and again


Activity 2: Dividing the Snack Equally

A bag contains carrot coins or m&m chocolates. You have a balance, but no weights. How would you give several kids an equal share of the snack? It’s okay to have some left over, as long as each kid gets the same amount.4 kids:

5 kids:

8 kids:

15 kids:


Using only Four Weights

Challenge: What are the best four weights to use, if we can place them on both sides of the balance?

Using 1, 2, 5, 12 gram weights,we can weigh any amount of material from 1 to 20 grams.

1, 2, 4, and 8 grams are obvious.3 = 2 + 1 5 = 4 + 16 = 4 + 2 7 = 4 + 2 + 19 = 8 + 1 10 = 8 + 211 = 8 + 2 + 1 12 = 8 + 413 = 8 + 4 + 1 14 = 8 + 4 + 215 = 8 + 4 + 2 + 1

A chemist has a balance and fixed reference weights of 1, 2, 4, and 8 grams. Show that she can weigh any amount of material from 1 to 15 grams by putting the fixed weights on one side and the material to be weighed on the other.


Activity 3: Weighing with Four Weights

12 = 3 – 1 34 = 3 + 1 5 = 9 – 3 – 1 6 = 6 – 3 7 = 9 – 3 + 18 = 9 – 1 9 10 = 9 + 1

16 =17 = 18 = 19 =20 = 21 =22 =23 =24 =25 =

31 =32 =33 =34 =35 =

26 =27 =28 =29 =30 =

11 =12 =13 =14 =15 =

36 =37 =38 =39 =40 =

Show how to weight any amount from 1 to 40 grams.

1

27grams

9grams

3grams


Find the Lighter Counterfeit CoinWe have 3 coins. Two are good coins; one is a counterfeit coin thatweighs less. Identify the counterfeit with one weighing on a balance.

We have 9 coins; eight good coins and a counterfeit coin that weighs less. Identify the counterfeit with 2 weighings.

Compare coins 1 & 2. If they weigh the same, coin 3 is counterfeit; otherwise the lighter of the two is counterfeit.


Activity 4: Heavier Counterfeit CoinWe have 3 coins. Two are good coins; one is a counterfeit coin thatweighs more. Identify the counterfeit with one weighing on a balance.

We have 9 coins; eight good coins and a counterfeit coin that weighs more. Identify the counterfeit with 2 weighings.

Challenge: Find one heavier counterfeit coin among 27 with 3 weighings on a balance.


Odd Coin: Lighter or HeavierWe have 12 coins. Eleven are good coins; one is a counterfeit thatweighs less or more than a good coin. Identify the counterfeit coin and its relative weight with a minimum number of weighings on a balance.

Hint: First do it for 3 coins, one of which is a counterfeit, using only two weighing,

If equal, 3 is the counterfeit. Weigh 3 against 1 to see if it is lighter or heavier.

If 1 lighter than 2, coin 3 must be good. Weigh 1 against 3. If equal, 2 is counterfeit & heavier. Otherwise, 1 is counterfeit & lighter.

Compare1 & 2

If 1 heavier than 2, do as in the previous case.


Odd Coin: Lighter or Heavier (cont.)

Hint: First divide the coins into three groups of 4 coins: A, B, and C.

We have 12 coins. Eleven are good coins; one is a counterfeit thatweighs less or more than a good coin. Identify the counterfeit coin and its relative weight with a minimum number of weighings on a balance.

CompareA & B

If A = B, then C contains the counterfeit coin.

Weigh 3 coins from C against 3 good coins.If equal, the lone remaining coin in C is counterfeit and one more weighing is enough to tell if it’s lighter or heavier than a good coin.

If the three C coins are lighter, then . . .

If the three C coins are heavier, then . . .

If A < B or A > B . . .


Measurement with Scales

Bathroom scales are usually not accurate for small weights (say, less than 30 lbs) or very large weights (say, more than 200 lbs). How would you weigh a baby who is about 20-25 lbs using a bathroom scale?

We have 100 oranges in a heavy cardboard box. How can you find the weight of the oranges (without the box) using a bathroom scale?

How would you weigh the 100 oranges accurately using the bathroom scale if you estimate the oranges to weigh 20-30 lbs and the cardboard box 3-5 lbs?

Can you accurately weigh a person who is about 250-300 lbs using two bathroom scales?


Activity 5: Measurement with a Scale

You have 5 packages of candy bars and a scale.Each candy bar is supposed to weigh 200 grams.One of the packages, however, contains candy bars that weigh 150 grams.Can you find the package with lighter candy bars using the scale only once?

Extra challenge: You only know that one package contains lighter candy bars but you don’t know the weight of regular or lighter bars. How many times do you need to use the scale to find out the two weights and the package with lighter bars?


Measuring LengthYou are curious about the height of a light pole in or next to your house. Can you measure its height without climbing it?

Hint: Think of shadows!

If a 1-foot upright ruler casts a 6-inch shadow and the shadow of the light pole is 10 ft, the pole is 20 ft high.

Standing next to a stream that you can’t cross, you and a friend make a bet. She says that the stream is 15 ft wide; you say that it is 20 ft wide. You have a 30-ft piece of string and a 1-ft ruler. How can you decide who is right?

Light pole

1-footruler


Measuring VolumeHow can we measure the volume of an irregular object such as a chess piece?

3. Measure the volume of the water that spills out by pouring it into measuring cups.

1. Put a container filled with water inside a larger container.

2. Slowly insert the item into the inner container.


Next LessonThursday, April 14, 2005

1. Think of a 2-digit number (10-99).2. Multiply your number by 5.3. Add 500 to the result of step 2.4. Add the number of your siblings.5. Double the result of step 4.

You got a 4-digit number that starts with 1, has your original 2-digit number in the middle, and double the number of your siblings at the end.

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Mar. 2005Measurement PuzzlesSlide 1 Measurement Puzzles A Lesson in the “Math + Fun!” Series